<< Chapter < Page Chapter >> Page >

Since Δ A = Δ x Δ y = Δ y Δ x , we can express d A as d x d y or d y d x . This means that, when we are using rectangular coordinates, the double integral over a region R denoted by R f ( x , y ) d A can be written as R f ( x , y ) d x d y or R f ( x , y ) d y d x .

Now let’s list some of the properties that can be helpful to compute double integrals.

Properties of double integrals

The properties of double integrals are very helpful when computing them or otherwise working with them. We list here six properties of double integrals. Properties 1 and 2 are referred to as the linearity of the integral, property 3 is the additivity of the integral, property 4 is the monotonicity of the integral, and property 5 is used to find the bounds of the integral. Property 6 is used if f ( x , y ) is a product of two functions g ( x ) and h ( y ) .

Properties of double integrals

Assume that the functions f ( x , y ) and g ( x , y ) are integrable over the rectangular region R ; S and T are subregions of R ; and assume that m and M are real numbers.

  1. The sum f ( x , y ) + g ( x , y ) is integrable and
    R [ f ( x , y ) + g ( x , y ) ] d A = R f ( x , y ) d A + R g ( x , y ) d A .
  2. If c is a constant, then c f ( x , y ) is integrable and
    R c f ( x , y ) d A = c R f ( x , y ) d A .
  3. If R = S T and S T = except an overlap on the boundaries, then
    R f ( x , y ) d A = S f ( x , y ) d A + T f ( x , y ) d A .
  4. If f ( x , y ) g ( x , y ) for ( x , y ) in R , then
    R f ( x , y ) d A = R g ( x , y ) d A .
  5. If m f ( x , y ) M , then
    m × A ( R ) R f ( x , y ) d A M × A ( R ) .
  6. In the case where f ( x , y ) can be factored as a product of a function g ( x ) of x only and a function h ( y ) of y only, then over the region R = { ( x , y ) | a x b , c y d } , the double integral can be written as
    R f ( x , y ) d A = ( a b g ( x ) d x ) ( c d h ( y ) d y ) .

These properties are used in the evaluation of double integrals, as we will see later. We will become skilled in using these properties once we become familiar with the computational tools of double integrals. So let’s get to that now.

Iterated integrals

So far, we have seen how to set up a double integral and how to obtain an approximate value for it. We can also imagine that evaluating double integrals by using the definition can be a very lengthy process if we choose larger values for m and n . Therefore, we need a practical and convenient technique for computing double integrals. In other words, we need to learn how to compute double integrals without employing the definition that uses limits and double sums.

The basic idea is that the evaluation becomes easier if we can break a double integral into single integrals by integrating first with respect to one variable and then with respect to the other. The key tool we need is called an iterated integral.


Assume a , b , c , and d are real numbers. We define an iterated integral    for a function f ( x , y ) over the rectangular region R = [ a , b ] × [ c , d ] as

  1. a b c d f ( x , y ) d y d x = a b [ c d f ( x , y ) d y ] d x

  2. c d a b f ( x , y ) d x d y = c d [ a b f ( x , y ) d x ] d y .

The notation a b [ c d f ( x , y ) d y ] d x means that we integrate f ( x , y ) with respect to y while holding x constant. Similarly, the notation c d [ a b f ( x , y ) d x ] d y means that we integrate f ( x , y ) with respect to x while holding y constant. The fact that double integrals can be split into iterated integrals is expressed in Fubini’s theorem. Think of this theorem as an essential tool for evaluating double integrals.

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?